{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","month":"04","publication_identifier":{"eissn":["1469-7750"],"issn":["0024-6107"]},"type":"journal_article","status":"public","external_id":{"arxiv":["2402.05851"]},"title":"A central limit theorem for the matching number of a sparse random graph","abstract":[{"lang":"eng","text":"In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse Erdős–Rényi random graph, in an influential paper pioneering the so-called differential equation method for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the subcritical and critical regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the supercritical regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called Karp–Sipser process to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a non-constructive result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph."}],"oa_version":"Published Version","ddc":["510"],"file":[{"creator":"dernst","relation":"main_file","success":1,"date_created":"2025-04-15T13:18:43Z","file_id":"19564","checksum":"69ce9feaf64e776b99f3afd1041b1b11","access_level":"open_access","content_type":"application/pdf","file_name":"2025_JourLondMathSoc_Glasgow.pdf","file_size":392208,"date_updated":"2025-04-15T13:18:43Z"}],"publication":"Journal of the London Mathematical Society","department":[{"_id":"MaKw"}],"date_published":"2025-04-01T00:00:00Z","article_number":"e70101","file_date_updated":"2025-04-15T13:18:43Z","volume":111,"project":[{"_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777","name":"Randomness and structure in combinatorics"}],"quality_controlled":"1","date_created":"2025-04-13T22:01:19Z","publisher":"Wiley","corr_author":"1","article_type":"original","language":[{"iso":"eng"}],"intvolume":"       111","date_updated":"2025-04-15T13:23:52Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","image":"/images/cc_by_nc.png","short":"CC BY-NC (4.0)","name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)"},"arxiv":1,"day":"01","acknowledgement":"We would like to thank Christina Goldschmidt and Eleonora Kreačić for insightful discussions and clarifications about their work in the thesis [26]. Matthew Kwan was supported by ERC Starting Grant ‘RANDSTRUCT’ No. 101076777. Ashwin Sah and Mehtaab Sawhney were supported by NSF Graduate Research Fellowship Program DGE-2141064. Ashwin Sah was supported by the PD Soros Fellowship.\r\nOpen access funding provided by Institute of Science and Technology Austria/KEMÖ.","article_processing_charge":"Yes (via OA deal)","publication_status":"published","OA_type":"hybrid","_id":"19554","has_accepted_license":"1","author":[{"full_name":"Glasgow, Margalit","first_name":"Margalit","last_name":"Glasgow"},{"last_name":"Kwan","orcid":"0000-0002-4003-7567","first_name":"Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","full_name":"Kwan, Matthew Alan"},{"first_name":"Ashwin","full_name":"Sah, Ashwin","last_name":"Sah"},{"last_name":"Sawhney","full_name":"Sawhney, Mehtaab","first_name":"Mehtaab"}],"scopus_import":"1","license":"https://creativecommons.org/licenses/by-nc/4.0/","oa":1,"issue":"4","doi":"10.1112/jlms.70101","OA_place":"publisher"}